Essential Norms of Weighted Composition Operators between Bloch-Type Spaces

“…The essential norm of the operator uC ϕ : B → B was studied in [5,10]. The composition operators, weighted composition operators and related operators on the Zygmund space were studied in [1, 2, 4, 6-9, 14, 15, 18, 19].…”

Section: Let U ∈ H(d)mentioning

confidence: 99%

Essential norm of weighted composition operators from H∞ to the Zygmund space

Hu¹,

Zhu²

2016

J. Nonlinear Sci. Appl.

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“…Essential norm of weighted composition operators on the Bergman space and Bloch type spaces was computed by Cuckovic and Zhao [3] and MacCluer and Zhao [9], respectively. The notation A ≈ B means that there is a positive constant C such that B/C ≤ A ≤ CB.…”

Section: Introductionmentioning

confidence: 99%

Generalized weighted composition operators on the Bergman space

Sharma

2011

Demonstratio Mathematica

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. In this paper, we characterize boundedness and compactness of D n ϕ,ψ on the Bergman space A 2 . We also compute the upper and lower bounds of essential norm of this operator on the Bergman space. IntroductionThroughout this paper, we denote by H(D), the space of holomorphic functions on the open unit disk D = {z ∈ C : |z| < 1} in the complex plane C. For z, w ∈ D, let β z (w) = (z − w)/(1 − zw) be the Möbius transformation of D which interchanges 0 and z. The nth derivative of β z is βπ r dr dθ be the normalized area measure on D. For 0 < p < ∞, let L p be the Lebesgue space which contains measurable functions f on D such thatthe weighted Bergman space with the norm defined asRecall that the Bergman space A 2 (D) = A 2 is a functional Hilbert space of analytic functions on D and K z (w) = 1/(1 − zw) 2 is the Bergman kernel. Moreover, See [6] and [23] for more details on Bergman spaces. Let ϕ and ψ be holomorphic maps on the open unit disk D such that ϕ(D) ⊂ D. For a non-negative integer n, we define a linear operator D n ϕ,ψ as follows:We call it generalized weighted composition operator, since it include many known operators. For example, if n = 0 and ψ ≡ 1, then we obtain the composition operator C ϕ induced by ϕ, defined asIf ψ = 1 and ϕ(z) = z, then D n ϕ,ψ = D n , the differentiation operator defined as D n f = f (n) . If n = 0, then we get the weighted composition operator ψC ϕ defined as, the product of multiplication and differentiation operator. We provide a unified way of treating these operators on the Bergman space A 2 . Composition and weighted composition operators have gained increasing attention during the last three decades, mainly due to the fact that they provide, just as, for example, Hankel and Toeplitz operators, ways and means to link classical function theory to functional analysis and operator theory.In fact, some of the well known conjectures can be linked to composition operators. Nordgren, Rosenthal and Wintrobe [11] have shown that the invariant subspace problem can be solved by classifying certain minimal invariant subspaces of certain composition operators on H 2 , whereas, Louis de Branges used composition operators to prove the Bieberbach conjecture. For general background on composition operators, we refer [2], [15] and references therein. Weighted composition operators also appear naturally in different contexts. For example, W. Smith in [20] showed that the Brennan's conjecture [1], which says that if G is a simply connected planar domain and g is a conformal map of G onto D, then Ì G |g ′ | p dA < ∞ holds for 4/3 < p < 4, is equivalent to the existence of self-maps of D that make certain weighted composition operators compact. The surjective isometries of Hardy and Bergman spaces are certain weighted composition operators (see

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Essential Norms of Weighted Composition Operators between Bloch-Type Spaces

Cited by 113 publications

References 8 publications

Essential norms of composition operators between Bloch type spaces

Essential norms of composition operators between Bloch type spaces

Essential norm of weighted composition operators from H∞ to the Zygmund space

Generalized weighted composition operators on the Bergman space

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